Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier–Legendre series

. The article is devoted to approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 3 by the method of multiple Fourier–Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence order 1.5 for Ito stochastic di ﬀ er-ential equations with multidimensional noncommutative noise. These numerical methods are based on the so-called Taylor–Ito and Taylor–Stratonovich expansions. We calculate the exact lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Stratonovich stochastic integrals. Thus, the computational cost for the implementation of numerical methods can be signiﬁcantly reduced.

1 Strong Taylor-Ito and Taylor-Stratonovich numerical schemes with convergence order 1.5 The relevance of the problem of numerical integration of Ito stochastic differential equations (SDEs) is explained by a wide range of their applications related to the construction of adequate mathematical models of dynamical systems of various physical nature under random perturbations.Among these models, we note models in financial mathematics, biology, epidemiology, medicine, aerospace industry, hydrology, seismology, geophysics, genetics, electrodynamics, chemical kinetics [1][2][3][4][5][6][7][8][9][10].Moreover, Ito SDEs are used to solve various mathematical problems such as signals filtering with random noises, stochastic optimal control, stochastic stability and bifurcation analysis, parameter estimation of stochastic systems [1,2,9].In addition, exact solutions of Ito SDEs are known in rare cases, and besides, knowing the exact solution of Ito SDE does not always allow us to simulate it numerically without using special numerical methods.
One of the effective approaches to the numerical integration of Ito SDEs is an approach based on the Taylor-Ito and Taylor-Stratonovich expansions [1,2,9,10].The important feature of these expansions is iterated Ito and Stratonovich stochastic integrals with respect to components of the multi-dimensional Wiener process.
The article continues the research [11] and is devoted to the development of an effective method of iterated Ito and Stratonovich stochastic integrals approximation based on generalized multiple Fourier series and proposed by the first author of the article in [10].More precisely, we consider mean-square approximations of iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process by the method of multiple Fourier-Legendre series.It should be noted that the iterated Stratonovich stochastic integrals are more convenient to use than the iterated Ito stochastic integrals due to simplicity of their approximations [1,2,9,10,[12][13][14][15][16][17][18][19][20][21][22] (compare formulas (18) and (19)).
Among advantages of the method based on generalized multiple Fourier series [10,12] over the methods [1,2,8,9,[13][14][15][16][17][18][19][20][21] of mean-square approximation of iterated Ito and Stratonovich stochastic integrals we note the following.The well-known approach based on the Karhunen-Loeve expansion of the Brownian bridge process [1,8] (also see [17]) leads to iterated application of the operation of limit transition.At the same time the operation of limit transition is implemented only once in theorem 1 (see below).This feature is more appropriate for the approximation.Moreover, the method [10,12] allows one to calculate the exact lengths of sequences of independent standard Gaussian random variables individually for different iterated Ito or Stratonovich stochastic integrals.Thus, the computational cost for the implementation of numerical methods for Ito SDEs can be significantly reduced.For more details see chapter 6 of monograph [12].
Let (Ω, F, P) be a complete probability space, let {F t , t ∈ [0, T ]} be a nondecreasing rightcontinuous family of σ-algebras of F and let w t be a standard m-dimensional Wiener process, which is F t -measurable for any t ∈ [0, T ].We assume that the components w (i)  t (i = 1, . . ., m) of this process are independent.Consider an Ito SDE in the integral form where ω ∈ Ω, a(x, t): x 0 and w t − w 0 are independent (t > 0).Let us consider the following differential operators where a (i) (x, t), a (i) (x, t) are i-th components of a(x, t), a(x, t) respectively and B (i j) (x, t) is the i j-th element of B(x, t).
Assume that a(x, t) and B(x, t) are enough smooth functions with respect to the variables x and t.Consider the partition {τ q } N q=0 such that 0 We will say [1] that a numerical scheme y τ q , q = 0, 1, . . ., N converges strongly with order γ > 0 at time moment T to the process x t , t ∈ [0, T ] if there exist a constant C > 0, which does not depend on ∆ N , and a δ > 0 such that M| x T − y T | ≤ C(∆ N ) γ for each ∆ N ∈ (0, δ).
is defined by formula (10), l.i.m. is a limit in the mean-square sense, s (i = 0, 1, . . ., m) are i.i.d.N(0, 1)-r.v.'s for various i or j if i 0, C j k ... j 1 is defined by formula (11), Note that a number of generalizations and modifications of theorem 1 can be found in [12]. Let Combining estimates ( 8) and ( 12) for p 1 = . . .= p k = p and k = 3, we obtain It is not difficult to see that the multiplier factor 3! on the left-hand side of inequality (13) leads to a significant increase of computational costs for approximation of iterated Ito stochastic integrals.The mentioned problem can be overcome if we calculate the meansquare approximation error E p k exactly.Theorem 2 [12].Suppose that the conditions of theorem 1 are fulfilled.Then where i 1 , . . ., i k = 1, . . ., m; expression � ( j 1 ,..., j k ) means the sum with respect to all possible permutations ( j 1 , . . ., j k ).At the same time if j r swapped with j q in the permutation ( j 1 , . . ., j k ), then i r swapped with i q in the permutation (i 1 , . . ., i k ); another notations are the same as in theorem 1.
For the further consideration, we note that Theorem 3 [12].Suppose that {φ j (x)} ∞ j=0 is a complete orthonormal system of Legendre polynomials or trigonometric functions in the space L 2 ([t, T ]).Then, for the iterated Stratonovich stochastic integral of third multiplicity I * (i 1 i 2 i 3 ) (000)T,t the following expansion that converges in the mean-square sense is valid, where another notations are the same as in theorem 1.

Approximation of iterated Ito and Stratonovich stochastic integrals using Legendre polynomials
Using theorems 1 and 3 and the complete orthonormal system of Legendre polynomials in the space L 2 ([t, T ]), we obtain the following formulas for numerical modeling of iterated Ito and Stratonovich stochastic integrals from numerical schemes ( 6) and ( 7) [10,12] where 1 A is the indicator of the set A; i 1 , . . ., i 3 = 1, . . ., m, C 000 j 3 j 2 j 1 = Let E (l 1 ...l k )p k be the left-hand side of equality (14) for inetgral (4) and E * (l 1 ...l k )p k be the analogous value for integral (5).

Main results
This section is devoted to the optimization of approximation procedures for iterated Stratonovich stochastic integrals, i.e. we discuss how to essentially minimize the values q and q 1 from the previous section for iterated Stratonovich stochastic integrals.
From theorem 2 we obtain (equality ( 15)) [11,12] Obviously, conditions ( 20)-( 24) do not contain the multiplier factors 2!, 3! in contrast to estimate (12).However, the number of the mentioned conditions is quite large, which is inconvenient for practice.In [11] we proposed and confirmed the hypothesis that all formulas ( 20)-( 24) can be replaced by formulas ( 20) and ( 21) in which we can suppose that i 1 , . . ., i 3 = 1, . . ., m.At that we have no noticeable loss of the mean-square approximation accuracy of iterated Ito stochastic integrals.In this article, we propose and confirm similar hypothesis for iterated Stratonovich stochastic integrals.

Conclusion
As we mentioned above, existing approaches [1,2,8,9,[13][14][15][16][17][18][19][20][21] to the mean-square approximation of iterated stochastic integrals do not allow to choose different numbers p (see theorem 2) for approximations of different iterated stochastic integrals with multiplicity k = 2, 3, . . .and exclude the possibility for obtaining of approximate and exact expressions similar to formulas ( 12) and ( 14).This leads to unnecessary terms usage in the expansions of iterated Ito and Stratonovich stochastic integrals and, as a consequence, to essential increase of computational costs for the implementation of numerical methods for Ito SDEs.In this article, we have optimized method based on theorems 1-3, which makes it possible to correctly choose the lengths of sequences of standard Gaussian random variables required for the approximation of iterated Stratonovich stochastic integrals.Thus, the computational costs for the implementation of numerical methods for Ito SDEs based on the Taylor-Stratonovich expansion are significantly reduced.The analogous optimization for iterated Ito stochastic integrals is carried out in [11].
On the base of the obtained results we can recommend the following conditions for correct choosing the minimal natural numbers q and q 1 : E * (00)q 2 ≤ C(T − t) 4 , E * (000)q 1 3 ≤ C(T − t) 4  (for the strong scheme (7) with order 1.5).Here the left-hand sides of the above inequalities are defined by relations (20) and (21), correspondingly; C is a constant from condition (9).

Table 4 .
Comparison of numbers q 1 and p 1